Zermelo-Fraenkel Research Articles

Categoricity-like Properties in the First Order Realm

By classical results of Dedekind and Zermelo, second order logic imposes categoricity features on Peano Arithmetic and Zermelo-Fraenkel set theory. However, we have known since Skolem’s anti-categoricity theorems that the first order formulations of Peano Arithmetic and Zermelo- Fraenkel set theory (i.e., PA and ZF) are not categorical. Here we investigate various categoricity-like properties (including tightness, solidity, and internal categoricity) that are exhibited by a distinguished class of first order theories that include PA and ZF, with the aim of understanding what is special about canonical foundational first order theories.

Intuitionistic sets and numbers: small set theory and Heyting arithmetic

AbstractIt has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. We present an intuitionistic theory of the hereditarily finite sets, and show that it is definitionally equivalent to Heyting Arithmetic , in a sense to be made precise. Our main target theory, the intuitionistic small set theory is remarkably simple, and intuitive. It has just one non-logical primitive, for membership, and three straightforward axioms plus one axiom scheme. We locate our theory within intuitionistic mathematics generally.

Every Countable Model of Arithmetic or Set Theory has a Pointwise-Definable End Extension

Abstract According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of set theory, in which every individual is definable without parameters, challenges this conclusion. In this article, I introduce a flexible new method for constructing pointwise-definable models of arithmetic and set theory, showing furthermore that every countable model of Zermelo-Fraenkel ZF set theory and of Peano arithmetic PA has a pointwise-definable end extension. In the arithmetic case, I use the universal algorithm and its Σ n generalizations to build a progressively elementary tower making any desired individual a n definable at each stage n, while preserving these definitions through to the limit model, which can thus be arranged to be pointwise definable. A similar method works in set theory, and one can moreover achieve V = L in the extension or indeed any other suitable theory holding in an inner model of the original model, thereby fulfilling the resurrection phenomenon. For example, every countable model of ZF with an inner model with a measurable cardinal has an end extension to a pointwise-definable model of ZFC + V = L[μ].

Independence Results for Finite Set Theories in Well-Founded Locally Finite Graphs

We consider all combinatorially possible systems corresponding to subsets of finite set theory (i.e., Zermelo-Fraenkel set theory without the axiom of infinity) and for each of them either provide a well-founded locally finite graph that is a model of that theory or show that this is impossible. To that end, we develop the technique of axiom closure of graphs.

The continuum hypothesis: Its independence from Zermelo-Fraenkel set theory and impact on mathematical foundations

The Continuum Hypothesis, originally posited by the pioneering mathematician Georg Cantor in the latter part of the 19th century, stands as a cornerstone inquiry in the realm of set theory. This paper embarks on a journey, delving into the rudiments of set theory, before tracing the evolutionary trajectory of the Continuum Hypothesis. Central to this exploration is the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) a foundational pillar in modern set theoretic studies. The core tenets of ZFC are dissected, shedding light on the seminal proofs presented by luminaries in the field that underline the unprovability of the CH within this axiomatic system. Beyond its mathematical intricacies, the paper underscores the profound philosophical and practical implications of the CH in both set theory and the broader mathematical landscape. In synthesizing these insights, a profound realization emerges: the inherent limitations in establishing the veracity of the Continuum Hypothesis within the confines of ZFC. This poignant revelation beckons deeper introspection into the foundational underpinnings of mathematics, stirring both intrigue and reflection amongst scholars and enthusiasts alike.

Delving into the continuum hypothesis: A thorough examination of set theory and the nuances of mathematical infinity

This essay delves deep into one of the most intriguing mathematical puzzles of all time: the Continuum Hypothesis. Beginning with a robust foundational exploration, it sheds light on the key concepts of cardinality and power sets, which are pivotal to the realm of set theory. These foundational ideas set the stage for a deeper investigation into the relationship that the Continuum Hypothesis shares with real numbers and natural numbers. Historically, the Continuum Hypothesis has tantalized mathematicians. This paper takes a journey through time, highlighting the various endeavors to either prove or refute this hypothesis. Some of the most brilliant minds have grappled with its complexities, leaving behind a rich tapestry of mathematical thought. Furthermore, a significant portion of our discussion is centered on situating the Continuum Hypothesis within the context of Zermelo-Fraenkel Set Theory (ZFC). The intricate interplay between the hypothesis and ZFC offers profound insights and raises thought-provoking questions about the very nature of mathematical truth.

Axiomatization multisets: a comparative analysis

A multiset, unlike the classical set, allows for multiple instances of its elements. In this paper, we present a comparative analysis of theories on multisets. In particular, we examine, the first-order two-sorted multiset theory MST, and the single-sorted multiset theory MS that employs the same sort for multiplicities and the set they support. The logical strengths and significance of some axioms presented in these theories are investigated. The theory MST contains a copy of the Zermelo-Fraenkel set theory with the axiom of choice (ZFC) but is independent of ZFC. The single-sorted multiset theory describes a stronger theory that mirrors the Zermelo-Fraenkel set theory (ZF) and is equiconsistent with ZF and antifoundation. The two-sorted multisettheory MST is a conservative extension of the classical set theory, making it a suitable theory to assume when dealing with studies that involve multisets.

Higher-order games with dependent types

In previous work on higher-order games, we accounted for finite games of unbounded length by working with continuous outcome functions, which carry implicit game trees. In this work we make such trees explicit. We use concepts from dependent type theory to capture history-dependent games, where the set of available moves at a given position in the game depends on the moves played up to that point. In particular, games are modelled by a W-type, which is essentially the same type used by Aczel to model constructive Zermelo-Frankel set theory (CZF). We have also implemented all our definitions, constructions, results and proofs in the dependently-typed programming language Agda, which, in particular, allows us to run concrete examples of computations of optimal strategies, that is, strategies in subgame perfect equillibrium.

A Model in Which Well-Orderings of the Reals First Appear at a Given Projective Level, Part III—The Case of Second-Order PA

A model of set theory ZFC is defined in our recent research, in which, for a given n≥3, (An) there exists a good lightface Δn1 well-ordering of the reals, but (Bn) no well-orderings of the reals (not necessarily good) exist in the previous class Δn−11. Therefore, the conjunction (An)∧(Bn) is consistent, modulo the consistency of ZFC itself. In this paper, we significantly clarify and strengthen this result. We prove the consistency of the conjunction (An)∧(Bn) for any given n≥3 on the basis of the consistency of PA2, second-order Peano arithmetic, which is a much weaker assumption than the consistency of ZFC used in the earlier result. This is a new result that may lead to further progress in studies of the projective hierarchy.

On Hausdorff operators in ZF$\mathsf {ZF}$

AbstractA Hausdorff space is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every with , , where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in , i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (): is equivalent to “For every set X, the Cantor cube is effectively Hausdorff”. This enhances the result of Howard, Keremedis, Rubin and Rubin [13] that is equivalent to “Hausdorff spaces are effectively Hausdorff” in . The Boolean Prime Ideal Theorem and the statement “For every infinite set X, the Stone space of the Boolean algebra is effectively Hausdorff” are mutually independent. In particular, the latter statement is not provable in . The Axiom of Choice for non‐empty subsets of () is equivalent to each of “Separable Hausdorff spaces are effectively Hausdorff” and “The Cantor cube is effectively Hausdorff”. The Principle of Dependent Choices in conjunction with the Axiom of Choice for continuum sized families of non‐empty subsets of does not imply the axiom of choice for partitions of . The latter independence result fills the gap in information in Howard and Rubin's book “Consequences of the Axiom of Choice”. The axiom of countable choice for non‐empty subsets of is equivalent to each of “Denumerable Hausdorff spaces are effectively Hausdorff”, “Denumerable T3 spaces are completely normal” and “Denumerable Tychonoff spaces are Urysohn”.

Choice and independence of premise rules in intuitionistic set theory

Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and Gödel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed under the corresponding rules for finite types over N. It is also shown that the existence property (or existential definability property) holds for statements of the form ∃yσφ(y), where the variable y ranges over objects of finite type σ. This applies in particular to CZF (Constructive Zermelo-Fraenkel set theory) and IZF (Intuitionistic Zermelo-Fraenkel set theory), two systems known not to have the general existence property. On the technical side, the paper uses a method that amalgamates generic realizability for set theory with truth, whereby the underlying partial combinatory algebra is required to contain all objects of finite type.

Polynomial-Time Axioms of Choice and Polynomial-Time Cardinality

There is no single canonical polynomial-time version of the Axiom of Choice (AC); several statements of AC that are equivalent in Zermelo-Fraenkel (ZF) set theory are already inequivalent from a constructive point of view, and are similarly inequivalent from a complexity-theoretic point of view. In this paper we show that many classical formulations of AC, when restricted to polynomial time in natural ways, are equivalent to standard complexity-theoretic hypotheses, including several that were of interest to Selman. This provides a unified view of these hypotheses, and we hope provides additional motivation for studying some of the lesser-known hypotheses that appear here. Additionally, because several classical forms of AC are formulated in terms of cardinals, we develop a theory of polynomial-time cardinality. Nerode & Remmel (Contemp. Math. 106, 1990 and Springer Lec. Notes Math. 1432, 1990) developed a related theory, but restricted to unary sets. Downey (Math. Reviews MR1071525) suggested that such a theory over larger alphabets could have interesting connections to more standard complexity questions, and we illustrate some of those connections here. The connections between AC, cardinality, and complexity questions also allow us to highlight some of Selman’s work. We hope this paper is more of a beginning than an end, introducing new concepts and raising many new questions, ripe for further research.

SOME CONSEQUENCES OF AND

AbstractStrong Turing Determinacy, or ${\mathrm {sTD}}$ , is the statement that for every set A of reals, if $\forall x\exists y\geq _T x (y\in A)$ , then there is a pointed set $P\subseteq A$ . We prove the following consequences of Turing Determinacy ( ${\mathrm {TD}}$ ) and ${\mathrm {sTD}}$ over ${\mathrm {ZF}}$ —the Zermelo–Fraenkel axiomatic set theory without the Axiom of Choice: (1) ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {wDC}_{\mathbb {R}}$ —a weaker version of $\mathrm {DC}_{\mathbb {R}}$ .(2) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every set of reals is measurable and has Baire property.(3) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every uncountable set of reals has a perfect subset.(4) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that for every set of reals A and every $\epsilon>0$ :(a)There is a closed set $F\subseteq A$ such that $\mathrm {Dim_H}(F)\geq \mathrm {Dim_H}(A)-\epsilon $ , where $\mathrm {Dim_H}$ is the Hausdorff dimension.(b)There is a closed set $F\subseteq A$ such that $\mathrm {Dim_P}(F)\geq \mathrm {Dim_P}(A)-\epsilon $ , where $\mathrm {Dim_P}$ is the packing dimension.

Ordinal analysis and the set existence property for intuitionistic set theories.

On account of being governed by constructive logic, intuitionistic theories [Formula: see text] often enjoy various existence properties. The most common is the numerical existence property (NEP). It entails that an existential theorem of [Formula: see text] of the form [Formula: see text] can be witnessed by a numeral [Formula: see text] such that [Formula: see text] proves [Formula: see text]. While NEP holds almost universally for natural intuitionistic set theories, the general existence property (EP), i.e. the property of a theory that for every existential theorem, a provably definable witness can be found, is known to fail for some prominent intuitionistic set theories such as Intuitionistic Zermelo-Fraenkel set theory (IZF) and constructive Zermelo-Fraenkel set theory (CZF). Both of these theories are formalized with collection rather than replacement as the latter is often difficult to apply in an intuitionistic context because of the uniqueness requirement. In light of this, one is clearly tempted to single out collection as the culprit that stymies the EP in such theories. Beeson stated the following open problem: 'Does any reasonable set theory with collection have the existence property? and added in proof: The problem is still open for IZF with only bounded separation.' (Beeson. 1985 Foundations of constructive mathematics, p. 203. Berlin, Germany: Springer.) In this article, it is shown that IZF with bounded separation, that is, separation for formulas in which only bounded quantifiers of the forms [Formula: see text] are allowed, indeed has the EP. Moreover, it is also shown that CZF with the exponentiation axiom in place of the subset collection axiom has the EP. Crucially, in both cases, the proof involves a detour through ordinal analyses of infinitary systems of intuitionistic set theory, i.e. advanced techniques from proof theory. This article is part of the theme issue 'Modern perspectives in Proof Theory'.

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

On one of Specker's theorems

The Specker theorem is considered, stating that if for some cardinal m there is no strictly intermediate cardinal between m and 2m (briefly CH(m)), and also between 2m and 22m (briefly CH(2m)), then the cardinal 2m is an aleph (briefly WO(2m)). Rejecting the second condition CH(2m) in Specker's theorem, a different condition of local bounded selection of elements from the family of bijections of the set X, |X|=m, to sets X⊔[0,α), 0<α<ℵ(m) (briefly AC2m(ℵ(m))) is specified, which is not only a sufficient condition for 2m=ℵ(m), which strengthens Specker's theorem, but also a necessary condition for it. More precisely, if m is an infinite cardinal, then the formula 2m=ℵ(m) holds if and only if the following formula CH(m)∧AC2m(ℵ(m)) is true. Here ℵ(m) is a Hartogs number. The following generalized Specker theorems are also proved:(CH(m+ℵ(m))∧CH(2m))⇒WO(2m),(CH(m)∧CH(m+ℵ)∧ℵ≥ℵ(m))⇒WO(2m) and(CH(m+ℵ)∧CH(2m)∧ℵ(m)<ℵ≤2m)⇒WO(2m). The following nontrivial theorem is proved: Let m be a cardinal such that the formula CH(2m) is true and Hartogs number ℵ(2m) is a regular cardinal. Then folmula WO(m) is true if and only if m2=m and for every alephs ℵ,ℵ′ such that 2m<2ℵ and 2ℵ′<2m the following inequalities m<ℵ and ℵ′<m are valid, respectively. An unsolved problem is still being discussed: is the formula CH(m)⇒WO(m) true? Solving this problem boils down to solving the question of the consistency of the formula CH(m)∧¬WO(m) with the axioms of Zermelo-Fraenkel set theory without the axioms of choice and regularity (briefly ZF). The problem of consistency and independence of formulas CH(m)∧¬WO(m) and CH(m)∧¬WO(2m) with the axioms of set theory ZF is also discussed. Their independence from these axioms ZF is proved. The consistency of the last formula is known and the question of the consistency of the first formula with axioms ZF remains unresolved yet.

Anti-classification results for groups acting freely on the line

We explore countable ordered Archimedean groups from the point of view of descriptive set theory. We introduce the space of Archimedean left-orderings Ar(G) for a given countable group G, and prove that the equivalence relation induced by the natural action of GL2(Q) on Ar(Q2) is not concretely classifiable. Then we analyze the isomorphism relation for countable ordered Archimedean groups, and pin its complexity in terms of the hierarchy of Hjorth, Kechris and Louveau [29]. In particular, we show that its potential class is not Π30. This topological constraint prevents classifying Archimedean groups using countable subsets of reals. We obtain analogous results for the bi-embeddability relation, and we consider similar problems for circularly ordered groups, and o-minimal structures such as ordered divisible Abelian groups, and real closed fields. Our proofs combine classical results on Archimedean groups, the theory of Borel equivalence relations, and analyzing definable sets in the basic Cohen model and other models of Zermelo-Fraenkel set theory without choice.

Sharon Berry.A Logical Foundation for Potentialist Set Theory

This book offers a foundation for mathematics grounded in a collection of axioms for logical possibility in a first-order language. The offered foundation is argued to have various epistemological benefits, in particular as regards our justification for believing certain axioms of set theory, and more generally as regards the ‘access’ problem for mathematical objects. The foundation for mathematics is provided in stages. First, Berry argues for a ‘potentialist’ interpretation of standard ZF set theory in terms of some axioms for logical possibility that are argued to ‘seem clearly true’. Then, she argues for a Carnap-inspired account of the semantics of ordinary mathematical language in support of the idea that all mathematical discourse as usually pursued can be interpreted faithfully in terms of the foundation provided by her potentialist set theory. The end result, it is argued, is something approximating a logicist foundation for mathematics: standard mathematical truths are interpreted in terms of a pure logic of logical possibility.

A formalised theorem in the partition calculus

A paper on ordinal partitions by Erdős and Milner [7] has been formalised using the proof assistant Isabelle/HOL, augmented with a library for Zermelo–Fraenkel set theory. The work is part of a project on formalising the partition calculus. The chosen material is particularly appropriate in view of the substantial corrections [8] later published by its authors, illustrating the potential value of formal verification.

Should Type Theory Replace Set Theory as the Foundation of Mathematics?

Mathematicians often consider Zermelo-Fraenkel Set Theory with Choice (ZFC) as the only foundation of Mathematics, and frequently don’t actually want to think much about foundations. We argue here that modern Type Theory, i.e. Homotopy Type Theory (HoTT), is a preferable and should be considered as an alternative.